Induced connection on an associated bundle via horizontals

Let $\pi:P\to M$ be a principal G-bundle with principal connection $\omega$ , and let $E=P\times_G F$ be the associated bundle for a left $G$ -space $F$ . Write $q:P\times F\to E$ for the quotient map and denote by $\pi_E:E\to M$ the bundle projection.

Construction (horizontal distribution on E). For $[p,u]\in E$ , define the horizontal subspace

H^E_{[p,u]} := dq_{(p,u)}(H_p \times \{0\}) \subset T_{[p,u]}E,

where $H_p=\ker(\omega_p)\subset T_pP$ is the horizontal subspace of $P$ and $0\in T_uF$ is the zero vector. This is well-defined (independent of the representative $(p,u)$ ) because $H$ is $G$ -equivariant and the quotient identifies $(pg,g^{-1}u)$ with $(p,u)$ .

Then $TE$ splits as

T_{[p,u]}E = H^E_{[p,u]} \oplus \ker(d\pi_E)_{[p,u]},

giving an Ehresmann connection on $E$ .

In the vector bundle case (fiber a vector space), this induced connection coincides with the usual notion of connection on a vector bundle.

Examples

For $E=P\times_G G$ with left multiplication, the induced connection recovers the original principal connection under the identification $E\cong P$ .
For $E=\mathrm{Ad}(P)$ , the construction produces a natural connection on the adjoint bundle used in gauge theory to differentiate gauge parameters.
If $P$ is trivial over $U$ and $A$ is the local gauge potential, the induced horizontals on $U\times F$ are determined by $A$ acting through the infinitesimal action of $\mathfrak g$ on $F$ .